c 1998 Society for Industrial and Applied Mathematics

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1 SIAM J MARIX ANAL APPL Vol 20, No 1, pp c 1998 Society for Industrial and Applied Mathematics POSIIVIY OF BLOCK RIDIAGONAL MARICES MARIN BOHNER AND ONDŘEJ DOŠLÝ Abstract his paper relates disconjugacy of linear Hamiltonian difference systems LHdS and hence positive definiteness of certain discrete quadratic functionals to positive definiteness of some bloc tridiagonal matrices associated with these systems and functionals As a special case of a Hamiltonian system, Sturm Liouville difference equations are considered, and analogous results are obtained for these important objects Key words linear Hamiltonian difference system, Sturm Liouville difference equation, bloc tridiagonal matrix, discrete quadratic functional AMS subject classifications 15A09, 15A63, 39A10, 39A12 PII S Introduction he aim of this paper is to relate disconjugacy of linear Hamiltonian difference systems LHdS H x A x +1 + B u, u C x +1 A u and hence positivity of the discrete quadratic functional F Fx, u u B u + x } +1C x +1 0 to positive definiteness of a certain bloc tridiagonal symmetric matrix associated with H and F Here A, B, C are sequences of real n n matrices such that 1 I A are invertible and B,C are symmetric for all N 0 o introduce our problem in more detail, we first recall the relation of disconjugacy of linear Hamiltonian systems both differential and difference to positivity of corresponding quadratic functionals and to solvability of the associated Riccati matrix equation A statement of this ind is usually called a Reid roundabout theorem Proposition 11 [4, 6, 7] Consider the linear Hamiltonian differential system H x Atx + Btu, u Ctx A tu, where A, B, C : I [a, b] R n n are continuous real n n matrix valued functions such that Bt,Ct are symmetric and Bt is positive semidefinite for all t [a, b], and suppose that this system is identically normal in I, ie, the only solution x, u of H for which x 0 on some nondegenerate subinterval of I is the trivial solution x, u 0, 0 hen the following statements are equivalent Received by the editors March 19, 1997; accepted for publication in revised form by U Helme March 9, 1998; published electronically September 23, San Diego State University, Department of Mathematics, 5500 Campanile Dr, San Diego, CA bohner@saturnsdsuedu he research of this author was supported by the Alexander von Humboldt Foundation Masary University, Department of Mathematics, Janáčovo Nám 2A, CZ Brno, Czech Republic dosly@mathmunicz he research of this author was supported by grant 201/96/0401 of the Czech Grant Agency 182

2 BLOCK RIDIAGONAL MARICES 183 i System H is disconjugate in I, ie, the 2n n matrix solution X U of H given by the initial condition Xa 0, Ua I has no focal point in I, ie, det Xt 0in a, b] ii he quadratic functional F b [ Fx, u u tbtut+x tctxt ] dt a is positive for every x, u : I R n satisfying x Atx + Btu, xa 0 xb and x 0 in I iii here exists a symmetric solution Q : I R n n of the Riccati matrix differential equation R Q + A tq + QAt+QBtQ Ct 0 In the last decade, a considerable effort has been made to find a discrete analogue of this statement; see [1] and the references given therein Finally, this problem was resolved in [2] and the discrete Roundabout heorem reads as follows Proposition 12 [1, 2] Assume 1 hen the following statements are equivalent i System H is disconjugate in the discrete interval J : [0,N] N 0, N N, ie, the 2n n matrix solution X U of H given by the initial condition X 0 0, U 0 I has no focal point in J : [0,N +1] N 0, ie, Ker X +1 Ker X and D X X +1 I A 1 B 0 Here Ker and stand for the ernel and the Moore Penrose generalized inverse of the matrix indicated, respectively, and the matrix inequality stands for nonnegative definiteness ii he discrete quadratic functional F is positive for every x, u :J R n satisfying x A x +1 + B u, x and x 0 in J iii here exist symmetric matrices Q : J R n n such that I + BQ are invertible, I + BQ 1 B 0 in J, and solve the discrete Riccati matrix difference equation R Q +1 C +I A Q I + B Q 1 I A 1 he main difference between continuous and discrete functionals F and F is that the space of x, u appearing in the discrete quadratic functional has finite dimension his suggests investigating positivity of F not only via oscillation properties of H and solvability of R as given in the roundabout theorem which are typical methods for an infinite-dimensional treatment, but also via linear algebra and matrix theory he main idea of this approach can be illustrated in the case of the Sturm-Liouville equation 2 r y +p y +1 0 and the corresponding quadratic functional 3 as follows J y r y 2 + p y+1} 2 0

3 184 MARIN BOHNER AND ONDŘEJ DOŠLÝ Expanding the differences in J, for any y y } N+1 0 satisfying 4 y 0 0+1, we have J y r + p y+1 2 2r y y +1 + r y} 2 0 y 1 β 0 r 1 r 1 β 1 rn 1 r N 1 β N 1 where β r + p + r +1 Hence J y > 0 for any nontrivial y R N+2 satisfying 4 iff the matrix L : β 0 r 1 r 1 β 1 rn 1 y 1, r N 1 β N 1 is positive definite From the elementary course of linear algebra it is nown that L is positive definite iff all its principal minors 1 β 0, 2 β 0 β 1 r 2 1,, N det L are positive On the other hand, by i of Proposition 12 we have J y > 0 for any nontrivial y satisfying 4 iff the solution ỹ of 2 given by the initial condition ỹ 0 0,ỹ 1 1 r 0 satisfies 5 ỹ +1 0 and δ : ỹ r ỹ +1 0, 0,,N Using Laplace s rule for computation of determinants, we have the formula 6 β 1 1 r Expanding the forward differences in 2 we have y +2 1 r +1 [β y +1 r y ] his recurrent formula, coupled with 6 and the initial condition ỹ 0 0,ỹ 1 1 r 0, gives ỹ 2 β 0 1, ỹ 3 1 [ β0 β 1 r 2 ] 2 1, r 1 r 0 r 1 r 0 r 2 r 1 r 0 r 2 r 1 r 0 and by induction ỹ +1 1 [ β 1 1 r 2 ] r r 2 r 1 r r r 2 r 1 r 0

4 BLOCK RIDIAGONAL MARICES 185 Consequently, 7 δ ỹ r ỹ +1 1, 0 : 1 Now, by the Jacobi diagonalization method, there exists an N N triangular matrix M such that M LM diagδ 1,,δ N } From the last identity one may easily see why the quantities δ come to play in the definition of disconjugacy of 2 In this paper we establish a similar identity relating the quadratic functional F and the matrices D from Proposition 12 his identity reveals why the matrices D appear in the definition of disconjugacy of H In particular, we find a bloc triangular matrix M such that M LM diagd 1,,D N }, where L in this identity is the matrix representing the functional F see heorem 22 in the next section and D are given in i of Proposition 12 he paper is organized as follows he next section is devoted to preliminary results We recall some basic properties of solutions of H, and we also show the relation between higher order Sturm Liouville difference equations and LHdS H he main results of the paper, the equivalence of positive definiteness of a bloc tridiagonal matrix to nonnegative definiteness of certain matrices constructed via solutions of H which reduce to δ in the scalar case are given in section 3 In the last section we deal with LHdS H which correspond to higher order Sturm Liouville equations; here the results of the previous section are simplified considerably he statements of section 4 complement results of [3, 5] 2 Preliminary results Subject to our general assumption 1 we consider a linear Hamiltonian difference system H and the corresponding discrete quadratic functional F defined by F Here, x x } N0 and u u } N0 are sequences of R n -vectors, and we say that such a pair x, u isadmissible on J provided that x A x +1 + B u for all J holds An x is called admissible on J if there exists u such that the pair x, u is admissible on J he functional F is then said to be positive definite and we write F > 0 whenever Fx, u > 0 for all on J admissible x, u with x and 0 holds hroughout the paper we denote by X, U the principal solution of H at 0, ie, the solution introduced in Proposition 12i Concerning Moore Penrose inverses we will need the following basic lemma which is proved, eg, in [2, Remar 2iii]

5 186 MARIN BOHNER AND ONDŘEJ DOŠLÝ Lemma 21 For any two matrices V and W we have KerV KerW iff W WV V iff W V VW It is the goal of this paper to relate the condition from Proposition 12i to a condition on certain bloc tridiagonal n n matrices of the form 0 S 1 L S 1 1 S 1, N 0, S 1 1 where we put, for N 0, 8 C +I A B I A +B +1 and S B I A Let L L N Our first result then is the following heorem 22 Let x, u be admissible on J with x hen we have Fx, u L Proof Let x, u be admissible on J so that u B u u B B B u x +1I A x B I A x +1 x holds for all J hen, if x ,wehave Fx, u x +1C x +1 + x +1I A x } B I A x +1 x } x +1 B +1 x +1 +2x S x +1 + x B x } x +1 x +1 +2x S x +1 x B x x +1 x +1 +2x } S x +1 x N+1 B N x 0 B 0 x 0 0 N 1 1 x +1 x +1 +2x S x +1 } + x 1 0 L, and hence our desired result is shown

6 BLOCK RIDIAGONAL MARICES 187 By introducing the space A : x x } N0 is admissible on J with x , we can write an immediate consequence of heorem 22 Corollary 23 F > 0 iff L > 0 on A Note that L > 0onA, ie, χ Lχ>0 for all χ A\0}, is equivalent to M LM 0 and KerM LM KerM whenever M is a matrix with ImM A In the next section we will present such a matrix M for the general Hamiltonian case, and in the last section we will give this matrix M and further results for the Sturm Liouville case A Sturm Liouville difference equation n } SL µ r µ µ y +n µ 0, N 0 µ0 with reals r µ,0 µ n, such that rn 0 for all N 0 is a special case of a linear Hamiltonian difference system We define, for all N 0, A 1, B 0, 1 0 r n 9 r 0 r 1 and C r n 1 hen assumption 1 is satisfied and the following result from [3, Lemma 4] holds Lemma 24 Suppose 9 hen x is admissible on J iff there exists a sequence y y } 0 N+n 1 of reals such that x y +n 1 y +n 2 2 y +n 3 n 1 y for all 0 N +1 holds, and in this case the functional defined by F taes the form 10 Fx, u n 0 ν0 r ν ν y +n ν } 2, where u is such that x, u is admissible We conclude this section with two auxiliary results where we use the notation introduced in Proposition 12 Lemma 25 Let à : I A 1 and D X X +1ÃB, N 0

7 188 MARIN BOHNER AND ONDŘEJ DOŠLÝ i Suppose 1 IfKerX +1 KerX, then D is symmetric and KerX +1 KerB Ã ii Suppose 9 hen we have for all 0 n 1 KerX +1 KerX, ranx +1 +1, and D 0 Proof While part i is shown in [2, Remar 2 ii], ii is the contents of [3, Lemma 4] Lemma 26 We have, for all N 0, } 11 X+1 X +1 X U + B X X S X +1 X+1S X Proof he calculation X+1 X +1 + X S X +1 + X+1S X X+1 U+1 I A } U +X+1I A B I A X +1 + X+1B +1 X +1 X B I A X +1 X+1I A B X X+1U +1 X + U B U +X + U B B X + B U +X+1B +1 X +1 X B X + B U X + U B B X } X U + X B X shows that formula 11 holds 3 he linear Hamiltonian difference system In this section we present a matrix M satisfying ImM A We then give a proof of the following crucial result heorem 31 If KerX +1 KerX holds on J, then we have M LM diag D 1,,D N } o further motivate our investigations, first we would lie to briefly consider the case that the matrices 12 B are invertible for all J hen we have A R Nn, and F > 0iffL > 0 hen, if X, U is the principal solution of H at 0 such that X are invertible for all 1 N + 1, and if we put F +1 D 1 S 1 D 2 S 2 D S D 2 S 2 D S 1 1 D S 1 I it is easy to show that, for all 0 N, 13 D 1 +1 S D S, L +1 F +1 D +1 for all 0 N, 0 1 I

8 BLOCK RIDIAGONAL MARICES 189 and 14 L 1 L F +1 D +1 F +1 heorem 32 Assume 12 We put L 0 : I and D 0 : 0 hen we have recursively, for 0, 1, 2,,N, L +1 > 0 L > 0 and D+1 : S D } 1 15 S > 0 Proof Of course 15 is obvious for 0 However, if 15 holds for some 0 N 1, then put R 0 S L R L +1 > 0 L > 0 and R L 1 R > 0 R However, by 14, R L 1 R S D S, where our D here are exactly the D because of 13 and D 0 0 his proves our desired assertion Now we turn our attention to the general case of an LHdS H, where we assume 1 Let us define D ij for 1 i j N matrices by hen we have D ij X i X j D j D ii X i X i D i X i X i X ix i+1ãib i X i X i+1ãib i D i and, by Lemma 21, if KerX j KerX i, D ij X i X j D j X i X j X jx j+1ãjb j X i X j+1ãjb j We now define, for any m J, anmn mn matrix M m by D 11 D 12 D 1m M m 0 D D mm and put M M N heorem 33 If KerX +1 KerX holds on J, then ImM A Proof We assume KerX +1 KerX on J First, let Aand put x n c 0 : 0, c +1 : c X +1ÃB U c u for N 0, where u u } N0 is such that x, u is admissible on J Let d : U c u for N 0 We have X 0 c 0 0x 0, and X c x for some J implies X +1 c +1 X +1 c X +1 X +1ÃB U c u ÃX + ÃB U c ÃB U c u ÃX c + ÃB u Ãx + ÃB u x +1

9 190 MARIN BOHNER AND ONDŘEJ DOŠLÝ because of Lemmas 25i and 21 Hence Next, we have for j J, so that X c x for all 0 N +1 D j d j X j X j+1ãjb j U j c j u j X j c j c j+1 X j c j D ij d j X i X j D jd j X i X j X j c j X i c j for all 0 i j N because of KerX j KerX i and Lemma 21 herefore D ij d j X i ji for all 0 i N so that Conversely, let with hen we have x n N ji c j X i c N+1 c i X i X N+1 X N+1c N+1 + X i c i x i M d 1 d N ImM ImM and put x We pic d 1,,d N R n x i I A 0 x 0 D ij d j for all 1 i N ji I A 0 X 1 X j D N jd j B 0 X j D jd j ImB 0, j1 I A N +1 D NN d N D NNd N ImB N, and, for J \N}, I A x +1 x I A j+1 I A X +1 X } B U N j+1 j1 X +1 X j D jd j j+1 X X j D jd j j X j D jd j D d ImB by Lemma 25i so that x is admissible on J and hence X j D jd j D d x n A

10 BLOCK RIDIAGONAL MARICES 191 Our next result directly yields heorem 31 heorem 34 If KerX +1 KerX holds on J, then M M m+1l m+1 M m+1 m L m M m 0 0 D m+1 Proof Let J and m J \N} We define n n matrices P and R by X 1 0 X 2 P and R 0 X S hen we have the recursions Mm P M m+1 m X m+1 D m+1 0 X m+1 X m+1 D m+1 Lm R and L m+1 m R m m Hence by putting M M 0, M m+1 Mm P m+1 X m+1 D m+1 herefore the matrix M m+1l m+1 M m+1 turns out to be M ml m+1 Mm M m L m+1 P m+1 X m+1 D m+1 D m+1 X m+1 Pm+1L m+1 Mm D m+1 X m+1 Pm+1L m+1 P m+1 X m+1 D m+1 M ml m M m Ω m X m+1 D m+1 D m+1 X m+1 Ω m D m+1 X m+1 Λ m X m+1 D m+1 with Ω m M ml m+1 P m+1 and Λ m P m+1l m+1 P m+1 Hence our result follows directly from ii and iii of the following lemma Lemma 35 If KerX +1 KerX holds on J, then we have, for all J \N}, i Λ X+1 U +1 + B +1 X +1; ii D +1 X +1 Λ X +1 D +1 D +1 ; iii Ω 0 Proof First of all we have, by formula 11 of Lemma 26, Λ 0 X 1 0 X 1 X 1 U 1 + B 1 X 1 and, if Λ 1 X U + B X already holds for some 1,,N 1}, again by applying formula 11 we have Λ P X +1 L R R P X +1 P L P + P R X +1 + X +1R P + X +1 X +1 X +1U +1 + B +1 X +1

11 192 MARIN BOHNER AND ONDŘEJ DOŠLÝ Hence i is shown, and by i, for all J \N}, D +1 X +1 Λ X +1 D +1 D +1 U +1 X +1 D +1 + D +1 B +1 D +1 X +1 X +2Ã+1B +1 U +1 X +1 D +1 + X +1 X +2Ã+1X +1 X +1 D +1 X +1 X +2 X +2X +1 D +1 X +1 X +1 D +1 D +1 taes care of part ii Finally, Ω 0 0, and if Ω 1 0 already holds for some 1,,N 1}, then Ω M 0 L R P R X +1 M L P + R X +1 M 1 0 D X L P P + S X +1 Ω 1 D X Λ 1 + D S X +1 0 D U + B X D B X 0 + B U because of part i Hence iii follows, and all our desired results are shown 4 he Sturm Liouville difference equation In this section we deal with the case where H and F correspond to a higher order Sturm Liouville equation SL and its corresponding quadratic functional 10 he special structure of the matrices A, B, C enables us to simplify the results of the previous section We start with an identity which plays the crucial role in the proof of the main results of this section Lemma 41 Let X, U be the principal solution of LHdS corresponding to SL If X +1 is nonsingular, then } D X X+1ÃB 1 det X diag 0,,0, r n det X +1 Proof Nonsingularity of X +1 implies that Ker X +1 Ker X, hence the matrix D X X+1ÃB 1 is symmetric according to Lemma 25i, and since B diag0,,0, 1/r n }, the only nonzero entry of D is in the right lower corner Using Laplace s rule, we have det X det [I A X +1 B U ] 0 0 det I A X U r n n,1 U r n n,n [ ] n I A X +1 n,ν 1 U n,ν adj [I A X +1 ] ν,n ν1 r n n I A X +1 B U n,ν [I A X +1 ] 1 det [I A X +1 ] ν,n ν1

12 n ν1 BLOCK RIDIAGONAL MARICES 193 X n,ν X+1Ã 1 deti A det X +1 ν,n det X +1 X X 1 +1Ã r n det X +1 D n,n, n,n r n det X +1 X X 1 +1ÃB n,n so the desired result follows In the next statement and its proof we suppose that the matrices X, L, M are the same as in the previous section and that the matrices A, B, C in H are given by 9 heorem 42 Suppose that X n,,x N+1 are nonsingular and denote 16 d : D n,n det X r n det X +1, n,,n hen there exists an nn N n +1 matri such that N LN diagd n,,d N } Proof Observe that the assumption of nonsingularity of X n,,x N+1 corresponds to the assumption Ker X +1 Ker X in heorem 31 because of Lemma 25 ii Denote by M [j] R nn, j 1,,nN, the columns of the matrix M and let N be the nn N n+1 matrix which results from M after omitting all zero columns, ie, N [ M [n2] M [nn+1] M [nn]] Since D 1 D n 1 0 for LHdS corresponding to SL by Lemma 25 ii, we have M LM diag0,,0,d n,,d N } by heorem 31 By Lemma 41, D j diag0,,0,d j }, j n,,n, and the statement follows from the relation between M and N Sturm Liouville equations may be investigated also directly, ie, not as a special case of LHdS Let t 1,,t n be n-dimensional vectors with entries µ 1 t µ ν 1 n ν, µ 1,,n, n ν with the usual convention that n 0if>nor <0, and denote by the n n matrix whose columns are the vectors t ν, ie, [t 1 t n ] Furthermore, let P p i,j bethenn N n + 1 matrix with n-vector entries p i,j R n given by p i,j t n+j i, with the convention that t l 0ifl<1orl>n It follows from Lemma 24 that if H corresponds to a Sturm Liouville equation SL, then x, u satisfying x is admissible iff there exists y y } N n such that x y ḳ y +n 1 ;

13 194 MARIN BOHNER AND ONDŘEJ DOŠLÝ hence x 1,,x N Aiff P y ṇ Consequently, if A, B, C are given by 9,,S by 8, and K : P LP, wehave Fy n 0 ν0 y ṇ r ν ν y +n ν 2 y ṇ P LP x +1 C x +1 + u } B u 0 y ṇ y ṇ K Expanding the differences in 10, it is easy to see that K isa2n + 1-diagonal matrix Our next computations extend the results of the first section by relating the quantities d from 16 in section 1 for n 1 denoted by δ to the principal minors of the matrix K First we loo for a relation between the nn N n+1 matrices P and N and the corresponding representation for x 1,,x N A By the previous considerations and heorem 33, x 1,,x N Aiff there exists c c n,,c N such that N c ṇ c N y ṇ P From the last equality we will find a relation between vectors c c n,,c N and y y n,, Note that these vectors are determined uniquely since the matrices P and N have full ran Denote by d j i the last column of the matrix X ix j D j aing into account the form of the matrices P and N, we get the system of equations t 1 c N d N N, 1 t 1 + t 2 c N 1 d N 1 N 1 + c nd N N 1 17, y n t 1 c n d n c N d N 1 From this system of equations, we see that there exists an upper triangular matrix B B i,j R N n+1 N n+1 such that y Bc and that this is just the matrix which reduces the 2n + 1-diagonal matrix K P LP to the diagonal form Indeed, we have Fy y ṇ c ṇ y ṇ K c ṇ N LN L c N c N

14 BLOCK RIDIAGONAL MARICES 195 y ṇ y ṇ B 1 diagd n,,d N }B 1 for any y y n,, R N n+1 ; hence B KB diagd n,,d N } Observe also that the diagonal entries of the matrix B are B, 1 n 1 d n+ 1, 1,,N n + 1 this follows directly from 17, and that KB j, B K,j diagd n,,d N }B 1,j 0 j 1,, 1, 1 n 1 j Here we used the information about diagonal entries of B he last expression may be regarded as a system of linear equations for B 1,,,B,, and by Cramer s rule we have B, 1 n 1 d n+ 1 1n 1 1 Now we can summarize our previous computations and relate the quantities d from 16, n,,n, to the principal minors of K his statement may be viewed as a direct extension of 7 to higher order Sturm Liouville difference equations Here, similarly as in the previous theorem, X, U is the principal solution of H with A, B, C given by 9, and the assumption of nonsingularity of the matrices X n,,x N+1 has the same meaning as in heorem 42 heorem 43 Suppose X n,,x N+1 are nonsingular and let be the principal minors of the matrix K hen, for all 1 N n +1, d n+ 1 1, 0 : 1 REFERENCES [1] C D Ahlbrandt and A C Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Academic Publishers, Boston, MA, 1996 [2] M Bohner, Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions, J Math Anal Appl, , pp [3] M Bohner, On disconjugacy for Sturm-Liouville difference equations, J Difference Equations and Appl, , pp [4] W A Coppel, Disconjugacy, Lecture Notes in Math 220, Springer-Verlag, Berlin, 1971 [5] O Došlý, Factorization of disconjugate higher order Sturm-Liouville difference operators, Comput Math Appl, to appear [6] W Reid, Ordinary Differential Equations, John Wiley, New Yor, 1971 [7] W Reid, Sturmian heory for Ordinary Differential Equations, Springer-Verlag, New Yor, 1980